Uniqueness result for Almost Periodic Distributions depending on time and space and an application to the unique continuation for the wave equation

Abstract

Let $\Omega \subset {\mathbb{R}}^N$, $N=1,2,3$, be an open bounded and connected set with continuous piecewise ${\mathrm{C}}^{\infty }$ boundary. Here we deal with almost periodic distributions of the form $u(t,x)={\sum }_{n=0}^{+\infty }{c}_n{S}_n(x){\mathrm{e}}^{i{\lambda }_{n}t}$ where $(c_n{)}_{n\in \mathbb{N}}\subset \mathbb{C}$ belong to the space of slowing growing sequences $s^{\prime }$, and $({\lambda }_n^2{)}_{n\in \mathbb{N}}\subset \mathbb{R}$ and $(S_n{)}_{n\in \mathbb{N}}\subset {\mathrm{H}}_0^1(\Omega )$ are respectively the eigenvalues and eigenvectors of the Laplacian. Given $\omega \subset \Omega$, we prove that there exists $T_{max}(\Omega ,\omega )>0$ depending only on $\Omega$ and $\omega$ such that if $T>T_{max}(\Omega ,\omega )$ and $u{|}_{\omega \times ]-T,T[}=0$, then $u\equiv 0$. Using this result we prove a unique continuation property for the wave equation.

11 pages, 3 figures